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10-2. Parabola, Ellipse, Hyperbola
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For $0<\theta<\pi / 2$, if the eccentricity of the hyperbola $\mathrm{x}^2-\mathrm{y}^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the value of $\theta$ is :
A
$\frac{\pi}{6}$
B
$\frac{5 \pi}{12}$
C
$\frac{\pi}{3}$
D
$\frac{\pi}{4}$
(JEE MAIN-2024)
Solution
$e_h=\sqrt{1+\sin ^2 \theta}$
$e_c=\sqrt{1-\sin ^2 \theta}$
$e_h=\sqrt{7} e_c$
$1+\sin ^2 \theta=7\left(1-\sin ^2 \theta\right)$
$1+\sin ^2 \theta=7\left(1-\sin ^2 \theta\right)$
$\sin ^2 \theta=\frac{6}{8}=\frac{3}{4}$
$\sin \theta=\frac{\sqrt{3}}{2}$
$\theta=\frac{\pi}{3}$
Standard 11
Mathematics
